Given that \(p = 1 + \sqrt{2}\) and \(q = 1 - \sqrt{2}\), we can evaluate the expression \(\frac{p^{2} - q^{2}}{2pq}\) as follows:

First, let's calculate the numerator

\(p^{2} - q^{2}\)

\(p^{2} - q^{2}= (1 + \sqrt{2})^2 - (1 - \sqrt{2})^2 \)

\( 1 + 2\sqrt{2} + 2 - (1 - 2\sqrt{2} + 2) \)

\(= 4\sqrt{2}\)

Now, let's calculate the denominator 2pq:

\(2pq= 2(1 + \sqrt{2})(1 - \sqrt{2}) \)

\(= 2(1 - (\sqrt{2})^2) = -2\)

Therefore, the value of the expression is:

\(\frac{p^{2} - q^{2}}{2pq} &= \frac{4\sqrt{2}}{-2} = -2\sqrt{2}